[Arrow] Integrate `ExtensionType::validate` within `ExtensionType::try_new`

**Is your feature request related to a problem or challenge? Please describe what you are trying to do.**

In #9677 @scovich and @alamb proposed integrating the new `validate` API within `try_new`.

It will reduce code duplication, improve modularity and prevent the potential slowdown from using the default value.

**Describe the solution you'd like**

Move the `ExtensionType` validation logic from the `try_new` into `validate`. Use `validate` in `try_new`.

**Describe alternatives you've considered**

Not do anything since there aren't too many plain validation cases in the codebase yet.

**Additional context**

Start by reading the ExtensionType::try_new and ExtensionType::validate entry points and compare their current validation responsibilities. Check existing ExtensionType validation cases and tests, then confirm that try_new uses validate without changing the resulting validation behavior or relying on a default value.

Written by the indexing model from the issue text.